The Li-Yau inequality and applications under a curvature-dimension condition

We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li-Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving new subsequents bounds on the heat kernel of the semigroup. Under positive curvature we moreover reach ultracontractive bounds by a direct and robust method.

[1]  M. Ledoux,et al.  Analysis and Geometry of Markov Diffusion Operators , 2013 .

[2]  A. Mondino,et al.  Li–Yau and Harnack type inequalities in RCD∗(K,N) metric measure spaces , 2013, 1306.0494.

[3]  N. Garofalo.  Curvature-dimension inequalities and Li-Yau inequalities in sub-Riemannian spaces , 2013, 1301.6581.

[4]  Paul W. Y. Lee.  Generalized Li-Yau estimates and Huisken's monotonicity formula , 2012, 1211.5559.

[5]  B. Qian.  A generalization of Hamilton’s gradient estimate☆ , 2012 .

[6]  Z. Qian,et al.  Sharp spectral gap and Li–Yau’s estimate on Alexandrov spaces , 2011, 1102.4159.

[7]  S. Varadhan.  On the behavior of the fundamental solution of the heat equation with variable coefficients , 2010 .

[8]  Feng-Yu Wang.  GRADIENT AND HARNACK INEQUALITIES ON NONCOMPACT MANIFOLDS WITH BOUNDARY , 2010 .

[9]  Fabrice Baudoin,et al.  Perelman’s Entropy and Doubling Property on Riemannian Manifolds , 2009, 0911.1819.

[10]  Xiangjin Xu,et al.  Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation☆ , 2009, 0901.3849.

[11]  C. Villani.  Optimal Transport: Old and New , 2008 .

[12]  Michel Ledoux,et al.  A logarithmic Sobolev form of the Li-Yau parabolic inequality , 2006 .

[13]  Qi S. Zhang,et al.  Sharp Gradient Estimate and Yau's Liouville Theorem for the Heat Equation on Noncompact Manifolds , 2005, math/0502079.

[14]  C. Villani.  Topics in Optimal Transportation , 2003 .

[15]  T. K. Carne.  HEAT KERNELS AND SPECTRAL THEORY: (Cambridge Tracts in Mathematics 92) , 1990 .

[16]  S. Yau,et al.  On the parabolic kernel of the Schrödinger operator , 1986 .

[17]  B. Qian.  Remarks on differential Harnack inequalities , 2014 .

[18]  Laboratoire de Probabilités et Modèles Aléatoires, , 2008 .

[19]  D. Bakry,et al.  Harnack inequalities on a manifold with positive or negative Ricci curvature , 1999 .

[20]  S. Yau.  Harnack inequality for non-self-adjoint evolution equations , 1995 .

[21]  S. Yau.  On the Harnack inequalities of partial differential equations , 1994 .

[22]  R. Hamilton.  Matrix Harnack estimate for the heat equation , 1993 .

[23]  E. Davies,et al.  Heat kernels and spectral theory , 1989 .